(ipcms, strasbourg) electron holography on magnetic ...esc.u-strasbg.fr › docs › 2012 ›...

• Magnetic properties of bulk materials– Dia-para-ferromagnetism– Susceptibility– Applications

• Magnetic properties of nano-objects– Moments and Curie temperature– Energies in competition– Applications

• Electron holography– Principle– Examples

Véronique Pierron-Bohnes(IPCMS, Strasbourg)

"Electron holography on Magnetic nanoparticles"

2

atomic moment in Fe clusters

MAGNETISM OF NANO-OBJECTS magnetic moments

IML Billas, JA Becker, A Chatelain, WA de Beer, PRL 71 (1993) 4067Magnetic moments of Iron clusters with 25 to 700 atoms and their dependence on temperature

G. Guzman-Ramırez, J. Robles, A. Vega, F. Aguilera-Granja, J. Chem. Phys. 134, 054101 (2011)Stability and magnetic phase diagram of ternary ferromagnetic3d-transition-metal nanoalloys with five and six atoms: DFT study

μat(μB) N=5 N=6 N=∞NiN 1.20 1.33 0.7CoN 2.60 2.33 1.7FeN 3.60 3.33 2.2

Density Functional Theory (DFT) calculations

atomic moment in Ni clustersS. E. Apsel, J. W. Emmert, J. Deng, and L. A. Bloomfield, Phys. Rev. Lett. 76, 1441 (1996), Surface-Enhanced Magnetism in Nickel Clusters

Extrapolated @ 0 K (to compensate thermal fluctuations)

Moments at 0K often larger on surface

bulk limit

The magnetic moments are sensitive to the presence of a surface : broken bonds and relaxation for example have a strong effect. In nanoparticles, the surface isnear to many atoms and the effect is very strong. Usually the moments are increased in small objects. It has been shown experimentally using a Stern and Gerlach experiment with iron and nickel clusters. The average atomic moment islarger than in bulk and varies with the configuration of the clusters. The calculation have been done only for very small numbers of atoms and show the same trend.

3

Curie temperature near a surface is often lower than in bulk (smaller coordination):

TC = αzJ/kB in Heisenberg model

⇒ M(T)/M(0K) smaller on surface⇒ decrease of magnetization on surface⇒ M(T) in nanoparticles cannot be predicted easily

MAGNETISM OF NANO-OBJECTS Curie temperature

U. Gradmann, in Handbook of MagneticMaterials, edited by K.H. J. Buschow (North-Holland, Amsterdam, 1993), Vol. 7, Chap 1.

Saturation magnetization of Ni(111) films on Re(0001) substrate versus T with differentnumbers of atomic layers

Curie temperature near a surface is often lower than in bulk

U. Gradmann, in Handbook of Magnetic Materials, edited by K.H. J. Buschow(North-Holland, Amsterdam, 1993), Vol. 7, Chap 1.

The coupling between atomic moments is also changed, giving rise to a change of the Curie temperature, that generally decreases due to the presence of a surface.The effect has been shown for example in thin film of nickel on a reniumsubstrate. The combination of the two effects : increase of the moment, but decrease of Curie temeprature imly that at room temperature either an increaseor a decrease of the magnetization can be observed.

∑−=ji

JE ijexch,

µµ 20)cos( MJF MMexch λμθ −=−=

and kBT !F = Fexch + Fani + Fdip +Fzee

)(2 rmA rr∇−= A=Ja2

Neighbour moments prefer to be aligned

at atomic scale macroscopicexpression

Mean Field approximation

Weiss

Simplest approximate expressions (continuous approximation)

MAGNETISM OF NANO-OBJECTS Energies in competition

The molecular field described by Weiss is the responsable of the magnetisationexistence locally. I will give the atomic scale and the macroscopic scaleexpressions for each energy. Weiss expression is at macroscopic scale. Magnetic moments of neighbours prefer to be aligned.

( )∑−=j

jjuani kE 2uµ )(cos2MKuani KF θ−=

∑−=ji

JE ijexch,



and kBT !F = Fexch + Fani + Fdip +Fzeeea

sy

easy

The interactions with the lattice (orbital moments) favour some directions (spin-orbit coupling)


Uniaxial anisotropy


iron

c

cobalt

KaniVeasy

diffi

cult

easy

When there is a coupling between the orbital moments and the spin moments, the total energy is sensitive to the orientation of the magnetisation compared to the cristal structure. There is a favored direction that is called the easy axis. In bcc Iron it is the [100] direction. In hcp Cobalt it is the direction of the c axis. An energy has to be given to the sample to deviate the magnetisation from the easydirection.

( )∑−=j


∑−=ji

JE ijexch,




Farer neighbour moments create a magnetic field

(magnetic dipole field)

HdM

mB

Magnetic MomentM = I S

( )[ ]∑ −⋅−=

i rij

iijijijdip 3

0 34

)(µuuµ

πμRH


Uniaxial anisotropy

for uniform magnetisation


( )[ ]∑

−−=

ji rE j

ij

iijijidip

,

3

4 30 µ

µuuµ

πμ

ddddip KMNMHF ≈=−= 200

22μμ


A third energy is due to the field created by the moments far from the farermoments. Depending on the symmetry of the sample: of the limits: sphere, cube, platelet… the average value is proportional to the squared magnetisation with a geometrical factor Nd called demagnetizing factor.

∑−=j

jzeeE Hµ0μ )cos(0 MHzee MHF θμ−=

( )∑−=j


∑−=ji

JE ijexch,




interactions with external magnetic field


Uniaxial anisotropy



( )[ ]∑

−−=

ji rE j

ij

iijijidip

,

3

4 30 µ

µuuµ

πμ


22μμ


There is also the classical interaction with an external fiel if any

Magnitude (Energy density) typical temperatureJ ~ 10+9 J/m3 1000K (TCurie)Ku ~ 10+5±2 J/m3 1mK-10KKd=½µ0M2 ~ 10+6 J/m3 1KkBT ~ 10+8 J/m3 300K

∑−=j

jzeeE Hµ0μ )cos(0 MHzee MHF θμ−=

( )∑−=j


( )[ ]∑

−−=

ji rE j

ij

iijijidip

,

3

4 30 µ

µuuµ

πμ


22μμ

∑−=ji

JE ijexch,

µµ 20)cos( MJF MMexch λμθ −=−= Mean Field approximation

Uniaxial anisotropy




Lecture of P.Panissod - French-Korean school NMS Strasbourg 2008



And the thermal energy that has to be compared to all other energies.The orders of magnetitude of these energies are quite different: the exchange energy is the largest, the Curie temperature gives its value.Magnetocrystalline and dipolar energies are quite simular and will be crucial to know the magnetic configuration.We will now see the different anisotropy origins in nano-objects

10

(a) magneto-crystalline anisotropy in bulk (related to structure symmetry)

Dependent on temperature

MAGNETISM OF NANO-OBJECTS Magnetic anisotropy

Anisotropy = F depends on magnetization direction for several reasons

HdR – R. Morel Grenoble 2009http://tel.archives-ouvertes.fr/docs/00/39/25/96/PDF/HDR_Morel.pdf

fcc Co hcp Co

Easyaxis [111] [0001]

K1(µeV/at)

K2(µeV/at)

K1(µeV/at)

K2(µeV/at)

300 K Paige 84 37 9.0

Weller 94 -3.9 0.90 30 9.0

Suzuki 94 -4.5 0.07

Fassbender 98 -5.9

77K Suzuki 94 -5.0 1.4

Paige 84 57 6.9

TCurie=1115°C=1388K

11

MS

θ

(b) Contribution of strains : F / V = Kε cos2θ for biaxial strains

Effect of magnetostriction


same order of magnitude as magneto-crystalline anisotropyin epitaxied films or nanoparticles

Co hcp

U. Gradmann, in Handbook of Magnetic Materials, edited by K.H. J. Buschow~North-Holland, Amsterdam, 1993!, Vol. 7, Chap 1. p26Le signe de l’anisotropie totale peut changer avec l’épaisseur selon le signe des différentes composantes

12

(c) Surface anisotropy (broken symmetry + cut bonds) Néel 1954F / V = F / Sd = σ/d with : ΣmL cos2(θm) où θm angle entre la liaison m et M

Bruno PRB 1989

In any surface, M tends to be perpendicular to surfaces d’après Gradmann

In surface anisotropy (110 for example)


Fe, Co, Ni : very thin films have easy magnetization axis perpendicular to surface(without strong strains)

Ni(111)2.2nm/Cu(111) U. Gradmann, in Handbook of Magnetic Materials, edited by K.H. J. Buschow ~North-Holland, Amsterdam, (1993), Vol. 7, Chap 1. p26

L. NéelL’anisotropie superficielle des substances ferromagnétiquesCR Acad.Sci. Paris 237 (1953) 1468J. Phys. Radium 15 (1954) 225

P. Bruno, PRB 39 (1989) 865Tight-binding approach to the orbital magneticmoment and magnetocrystalline anisotropy in transition metal alloys

U. Gradmann, in Handbook of Magnetic Materials, edited by K.H. J. Buschow~North-Holland, Amsterdam, 1993!, Vol. 7, Chap 1. p26Le signe de l’anisotropie totale peut changer avec l’épaisseur selon le signe des différentes composantesNéel a développé un modèle basé sur les liaisons coupées, modèle qui donne qualitativement les bons ordres de grandeur et le fait que les surfaces ouvertes ont un plus grande anisotropie de surface. La contribution de chaque liaison dépend du cos de l’angle de la liaison avec l’aimantation. Bruno a repris le modèle en liaisons forte en traitant le couplage spin-orbite en perturbations et retrouve plus quantitativement les même phénomènes.

13

S. Rusponi, T. Cren, N. Weiss, M. Epple, P. Buluschek, L. Claude, H. BruneThe remarkable difference between surface and step atoms in the magnetic anisotropy of two-dimensional nanostructuresNature Materials, 2, 546 (2003).


(d) Contribution of edge atoms → flat islands have much higher anisotropy:

Co/Pt(111)

Co/Pt/Pt(111) = Pt center/Co edges Edge atoms have an anisotropy per atom20 times higher than center atoms

S. Rusponi, T. Cren, N. Weiss, M. Epple, P. Buluschek, L. Claude and H. Brune, Nature Materials, 2, 546 (2003). The remarkable difference between surface and step atoms in the magnetic anisotropy of two-dimensional nanostructuresA basse T le modèle d’Ising (seulement une direction et deux sens) est plus proche de la simulation numérique qui rend tout en compte. Plus T augmente plus on va vers un modèle de Langevin (toutes les directions possibles)Formules valables seulement pour les articules épitaxiées étudiées selon leur axe de facile aimantation

14

diagonal with: Nxx+Nyy+Nzz=1

ellipsoïds

N=1/3N=0

N=1/2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Na

Nb = Nc

c/a

a > b = c

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Nc

Na = Nb

c/a

a = b > c

Scan Kneller

Hd = - N M ⇒

N tensor: coefficients of demagnetising field

JA Osborn, phys. Rev.67 (1945) 35ou E. Kneller, « Feromagnetismus » Springer Verlag, 1962

(e) Shape anisotropy (magnetostatic – dipolar origin: Fdip)

F / V = (μ0NMS2/2) cos2θ

Tendancies:Formation of several domains with magnetization parallel to surfacemagnetization along the largest dimension if monodomain

MS

θ

needle


JA Osborn, phys. Rev.67 (1945) 35Ou E. Kneller, « Feromagnetismus » Springer Verlag, 1962

15

2 domains ?

Hd2

Hs1Hd1

Hs2HdM

1 domain or

E=FdipV E=F'dipV+FwallSEdip= NdKda3 Edip= NdKda3

Eexch = 0 Eexch =2Aa

Single domain if NdKda3<4Aa

i.e.

21

21

21

⎟⎠⎞

⎜⎝⎛ +

ddd KA

KNAa ≈<

4

Scale #1: lexch=

Typical value : 3 nm(Kd=10+6 J/m3, A=10−11 J/m)(1000 atoms in the cube)

• Sets the boundary betweenatomic and continuous

• Sets the mesh size inmicromagnetic calculations

dKA

MAGNETISM OF NANO-OBJECTS Magnetic configurations of nano-objects

Eani = 0: Competition betweendipolar and exchange energies

abrupt domain wall

Lecture of P.Panissod - French-Korean school NMS Strasbourg 2008

single domain?

or two domains ?


16

n→∞ ⇒ Eexch=Aπ2/n→ 0 but Eani=nKu/2→∞

Eani=0 but Eexch=A


Abrupt domain wall ?or extented domain wall ?

Competition between dipolar, exchange and anisotropy energies


17

Eani=0 but Eexch=A

Scale # 2 : δwallTypical values (Kd=10+6 J/m3, A=10−11 J/m)

Soft material (Ku=10+3 J/m3) : δwall ~ 300 nmMedium (Ku=10+5 J/m3) : δwall ~ 30 nm

Hard material (Ku=10+7 J/m3) : δwall ~ 3 nm

uKA /π=

Energy and domain wall widthApproximation : dθ/dx=π/na, δwall= naEexch=Jna(Δθ2/2) =Ja2π2/2na =Aπ2/2δwall

Eani= Kuna(1/2) = Kuδwall/2Ewall minimum for

Eexch = Eani →

uwall KA /πδ =

uwall AKπσ =

σ per wall area unit


Eexch=Aπ2/n→ 0 but Eani=nKu/2→∞

18

HdM

s

s

s

1 domain 2 domains

Hd2

Hs1Hd1

Hs2

Scale # 3 : dsingTypical values(Kd=10+6 J/m3, A=10−11 J/m)

Soft material : dsing ~ 2 nm(Ku=10+3 J/m3)

Medium : dsing ~ 20 nm(Ku=10+5 J/m3)

Hard material : dsing ~ 200 nm(Ku=10+7 J/m3)

du KAK /6π=

E=FdipV E=F'dipV+FwallSEdip= NdKds3 Edip= (1/2) NdKds3

Eexch+Eani = 0 Eexch+Eani=

Single domain if NdKds3<2

i.e. s <

2sAKuπ

d

u

d KAK

Nπ2


Inconsistency of the description/evaluation for soft materialsdwall ~ 300 nm >> dsing ~ 2 nm ! -> more complicated configurations in soft materials: flower, vortex…

19

for cubes:Magnetic states of smallcubic particles with uniaxialanisotropyW. Rave, K. Fabian, A. Hubert Journal of Magnetism and Magnetic Materials, 190 (1998) 332


Magnetic states of small cubic particles with uniaxial anisotropyW. Rave, K. Fabian, A. Hubert Journal of Magnetism and Magnetic Materials, 190 (1998) 332

20

PhD Thesis Liliana Buda – Strasbourg


for discs:

Micromagnetic simulations of magnetisation in circular cobalt dotsComp. Mat. Sc. 24 (2002) 181 L. D. Buda, I. L. Prejbeanu, U. Ebels, K. Ounadjela

sans anisotropie KU avec anisotropie KU

Correlated Magnetic Vortex Chains in Mesoscopic Cobalt Dot ArraysM. Natali, I. L. Prejbeanu, A. Lebib, L. D. Buda, K. Ounadjela, and Y. Chen1,PRL 88 (2002) 157203

Investigation of 3D micromagnetic configurations in circular nanoelementsJournal of Magnetism and Magnetic Materials, Volumes 242-245, Part 2, April 2002, Pages 996-998L. D. Buda, I. L. Prejbeanu, U. Ebels, K. Ounadjela

Micromagnetic simulations of magnetisation in circular cobalt dotsCOMPUTATIONAL MATERIALS SCIENCE 24 (2002) 181 L. D. Buda, I. L. Prejbeanu, U. Ebels, K. Ounadjela

MAGNETISM OF NANO-OBJECTS Superparamagnetism

and kBT !F = Fexch + Fani + Fdip +FzeeCompetition between

exchange and thermal energies

Curie temperature

multidomain

energy coststo create a wall

< magnetostatic energy

superparamagnetism

thermal energy kT

>anisotropy energy

barrier Kan V

∼1 nm


blocked single-domain

Dc Kanis V ∝ kT

Magnetization in the easy direction

KaniV

Progressive change: position and visibilitydepend on the measurement

and kBT !F = Fexch + Fani + Fdip +FzeeCompetition between anisotropy

and thermal energies

Extrait du cours de G.Pourroy à l’école F-Corée NMS Strasbourg 2008

individual moment fluctuateswith τ < measurement time

Progressive change whose position and visibility depend on the measurement

Tb TC

superparamagnetismblockedferromagnetism paramagnetism

For a given size, at increasing temperature, the nanoparticle individual moments fluctuate

with characteristic time τ and disappears at TCurie

relaxation time τ=τ0 exp(Kani V/kT)

kTb ∝ Kanis V

Zero field cooled

Field cooled

H


kTb ∝ Jµ2

Extrait du cours de G.Pourroy à l’école F-Corée NMS Strasbourg 2008

How to image the magnetic domains ?

Cubic symmetry(Fe : bcc)

F. Bitter, Phys. Rev. 38, 1903 (1931) On Inhomogeneities in the Magnetization of

Ferromagnetic Materials

Kittel, C.: Physique de l’état solide. Ed. Dunod (1995)

What direction? Single or multidomains? Coupling between particles?Bulk samples: optical microscopy + colloidal suspension of ferro µP: resolution: 0.1mm

A single easydirection (Co : hcp) a1

a2

a3

c


micrometric samples:magnetic force microscopy : – sensitive to vertical magnetic field near the surface (stray field)

– resolution: 50 nm

magnetism.eu/esm/2003-brasov/slides/thiaville-slides-2.pdf

Image avec champs de fuite

25


micrometric samples:magnetic force microscopy : – sensitive to vertical magnetic field near the surface (stray field)

– resolution: 50 nm

magnetism.eu/esm/2003-brasov/slides/thiaville-slides-2.pdf

26

27

Interaction between dots: mainly dipolar interactionsExplain strange hysteresis loops

MAGNETISM OF NANO-OBJECTS Interactions and distributions

M. Natali, I. L. Prejbeanu, A. Lebib, L. D. Buda, K. Ounadjela, and Y. ChenPhys. Rev. Lett. 88, 157203 (2002) Correlated Magnetic Vortex Chains in Mesoscopic Cobalt Dot Arrays

The influence of the dipolar interactions is reflected in both the nucleation and annihilation field’s behavior. Their variation with the interdot separation, for an array of dots with f 550 nm, is illustrated in Fig. 1(c) and is compatible with a significant increase of the interaction fields for spacing below 1000 nm. The experimental data are well fitted by the expression Hn(a) = Hn(a)infini+/- 4.2Ms (V/S**3) that describes the interaction field in a twodimensional square lattice of dipoles. Here V denotes the volume of the dot and Ms is the saturation magnetization of Co (1400 emu/cm**3). The fitting parameter used, Hn(a)infini, represents the nucleation/annihilation field for isolated dots. The right-hand sideterm gives the dipolar interaction field which acts at nucleation (1) and annihilation (2).M. Natali, I. L. Prejbeanu, A. Lebib, L. D. Buda, K. Ounadjela, and Y. ChenPhys. Rev. Lett. 88, 157203 (2002) Correlated Magnetic Vortex Chains in Mesoscopic Cobalt Dot Arrays

How to image the magnetic domains ?What direction? Single or multidomains? Coupling between nanoparticles?Bulk samples: optical microscopy + colloidal suspension of ferro µP

nanometric samples:Electron holography : sensitive to the phase changes of the electronic wave – resolution: 5 nm

Other methods:

Spin polarised STM (lecture of W. Wulfhekel)

Ballistic Electron Magnetic Microscopy…

Applications of nano-magnetism

A magnet can be oriented in a field

A magnet produces a field

A magnet has a remanence

Electric resistivity of a magnetic system depends on the magnetisation orientation

A magnet can be heated using hysteresiscycling

Nanomotors, drug delivery, bacteria, pigeons…

Reading heads

High density information storage

GMR reading heads, sensors

Hyperthermia (cancer healing)


A magnet can be oriented in a fieldNanomotors, drug delivery, bacteria, pigeons…

http://www.sciencedirect.com/science/article/pii/S1748013207700841

A magnet can be heated using hysteresiscycling Hyperthermia (cancer healing)


A magnet can be oriented in a fieldNanomotors, drug delivery, bacteria, pigeons…

magnetic resonance imaging (MRI)

http://www.sciencedirect.com/science/article/pii/S1748013207700841

A magnet can be heated using hysteresiscycling Hyperthermia (cancer healing)


A magnet produces a fieldReading heads

5 μmMFM image of bits magnetized in the hard drive

M

MA magnet has a remanence

High density information storage




http://news.cnet.com/2300-1041_3-6213399-2.htmlHitachi new GMR head – 50 nm tracks

(ipcms, strasbourg) electron holography on magnetic ...esc.u-strasbg.fr › docs › 2012 ›...

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